Fourier expansion of a normalized Eisenstein series

For even

k \geq 4

, the normalized level-one Eisenstein series has expansion

E_{k} (τ) = 1 + \frac{2}{ζ ( 1 - k )} \sum_{n \geq 1} σ_{k - 1} (n) q^{n} = 1 - \frac{2 k}{B _{k}} \sum_{n \geq 1} σ_{k - 1} (n) q^{n} .

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- Divisor-sum convolution identity of weights four and eight Fourier expansion of a normalized Eisenstein series

Divisor-sum convolution identity of weights four and eight

The identity

E_{8} = E_{4}^{2}

gives

σ_{7} (n) = σ_{3} (n) + 120 \sum_{i = 1}^{n - 1} σ_{3} (i) σ_{3} (n - i) .

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 1 / a / Solution

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